Readings Reflections
The readings reflections have two main purposes:
1) to hold you accountable for careful reading of and reflection on the readings assigned in class; and
2) to provide you with a record of what you've learned and thought about as a result of the readings.
The readings reflections will be evaluated using the following criteria:
completeness and timeliness of the entries;
comprehension of the main ideas of the readings; and
depth and quality of integration of the ideas with your own thinking.
Submit your readings reflection before reading anyone's on the Wiki page and then paste it into the existing reflection page for that current reading. This is due by Sun 10/9 by 11pm.
The article, "Why, Why Should I Justify?", discusses different student approaches for the Beam Design problem, which incoporates four NCTM Principles and Standards. It can be challenging for students to justify their answers, so it is up to the teacher to create an environment condusive to these types of problems. I believe the teacher has to establish these norms early on in the school year. The longer students go without justifying their answers, the harder it is going to be for them to justify later in life. I am a product of this. For most of my schooling, I was never required to justify my generalizations, now it is difficult to get in that mindset. In the article, it mentions starting as early as fifth grade (440-3). So as middle school teachers, we should strongly encourage our students to validate their generalizations.
The four categories of justification are as follows: no justification/procedural justification, empirical justification, generic examples, and deductive justification. The goal is for students to perform deductive justification, which means to show how the rule was derived and be able to explain how it works. Proper justifications will apply to all cases, not just generic examples. Requiring justification in class also requires students to be confident. Students need their generalizations to work for all cases. Thus, they may need to try more than one approach to get a justification that works. If a student lacks mathematical confidence, they will be less likely to take the time to find a generalization that applies to all cases. This also must be encouraged at a young grade.
Students that are able to justify not only help themselves, but help their peers to understand the situation. Teachers need to encourage justification discussions and tasks that focus these discussions. The more we implement activities, such as the Beam Design, the better off our students will be. They will develop problem solving skills that will assist them not just in math class, but in the real world.
Hailey McDonell
The article “Why, Why Should I Justify?” explored the idea of students learning how to justify the solutions they are coming up with. As I read the article, I couldn’t help but notice how much I identified with the students at times. At the same time, the role of the teacher discussed and the way the classroom should be run seems to not only be practical but also quite beneficial for the students. Justification is pertinent to explaining how you came to a solution because it provides validity to your solution. Additionally, it helps convince others your explanation is valid (3-2). A weak justification, or none at all, leaves those who are trying to see how you obtained a certain solution confused and not convinced. The article stated there four types of broad justifications: (1) no justification or procedural only, (2) empirical, (3) generic examples, and (4) deductive justification (3-3). Many times, I believe, teachers allow their students to believe procedural justification is okay and works as means of an explanation. As we saw when we saw when we watched Cathy’s video, one student’s justification was “flip the fraction and multiply”. This does not justify anything mathematically. This may be the step the student takes to get to an answer but that doesn’t mean it proves anything. Additionally, from personal experience, students believe examples prove it all. I know when I was in the proofs course last semester it was engrained in my head to not assume if it worked for some it worked for all. There may always be that counter example. However, coming from a middle school mind, I can see how examples may seem like justification. It seems to be showing why something works. It is our job as teachers to show them why it is not justification. We can do this by asking the right questions. The questions asked by the teachers seem to be most important piece in changing the student’s way of thinking about justification. There are two things teachers do to help students think in the way of correct justification: (1) creating tasks which focus on justification and (2) encouraging justification through discussion and questions (4-5). As we have read before, asking questions forces the students to think for themselves. It puts the thinking back on them. I know as someone who is currently a student, when a teacher lectures and tells answers, acknowledges my answer as correct, I shut down. I know this is true for many if not all other students. If I know I have the right answer, what matters what the reasoning is. It seems to be the role of the teacher to make this different. The idea of group discussion and connecting the different techniques of doing a problem seems to be something that is very beneficial to students. First off the can see how others approached the problem and, hopefully, be able to connect their own ideas to that of others. More importantly, they will be able to test the validity of their own as well as others’ justification (6-1). This is a way for them to test if they know and have effective reasoning. This is something I feel I didn’t have the chance to do very much when I was in school and I feel if I did, I would have been able to examine more closely the reasoning behind my own work in school. Katey Cook
As students make the leap from concrete numerical procedures to more abstract algebra with variables, we, as teachers, must continue to push our students to justify their reasoning behind their mathematical ideas. These initial algebraic problems, like the Beam Design task, may look intimidating because of the new concept of a variable, but they are essential in bridging the gap between concrete and abstract thinking. By asking for justification of simple algebraic models, we allow students to see how a rule applies across various cases, construct generalizations to similar situations, and reflect on their own reasoning of their rules (4-2). This will promote the recognition of patterns for our students, who will become more adept at creating algebraic rules as they learn to assess their own reasoning.
As teachers, we must recognize which kind of justification is sufficient and guide students to provide this kind of justification for their ideas. Lannin, Barker, and Townsend classify justification using the subgroups of Simon and Blume (3-4). They are: (1) no/procedural justification, this would be characterized by statements like “I don’t know, just because,” as the student is unable to provide answers to why or how the rule works (2) Empirical justification, which would be the justification a student has if he or she is able to consider multiple cases, but has not provided reasoning for why the rule works over all of n (3) Generic examples are those in which students choose n to be some number, and explain their rule in terms of this one variable rather than in terms of any n (4) Deductive justification, this is the kind of justification that we must promote in our classrooms. Deductive justification explains why and how the rule works for all of n. These are the kinds of justifications that convince the classroom that the student’s statement is valid.
Though a student who provides a (1) justification may be aware that he or she is in need of help, students who provide type (2) and (3) justifications may feel more confident. They perhaps feel that they have reasoned out the answer because it works for random cases because they are more comfortable thinking in terms of concrete numbers rather than abstract variables. With these students, I feel that it is important to let them know that testing one’s rule on sample cases is a good idea to check one’s answer, but it is not how we can create a rule that works for all of n. This is why the article encourages questions from the teacher like, “How do you know your rule will always work?” and “How does your rule relate to the problem situation?” (5-3). These types of focusing questions will get students thinking in terms of any variable n rather than in terms of specific cases, n=5, n=8, etc.
Though the subject matter has changed from concrete numbers to abstract variables, it is important that we, as teachers, demand justification from our students so that they can continue to expand their reasoning capacities.
Valerie Gipper
The article, “Why, Why Should I Justify?”, discusses the importance of teaching students to not just conclude that an equation works for a variety of examples, but why the equation makes sense. The article goes on to discuss the four categories that student justifications generally fall into (440-4). We then see the three students explanations from the beginning of the article explained with regards to these four categories (440- 4). It seems that the fourth category (deductive justification) is the highest level of justification. In this level of justification the students reasoning applies to all situations and not just specific examples like in the other three cases (440-7).
Next the article talks about why teaching students to justify is important (441-1), which includes three reasons. I think it is very important for students to learn how to justify why things work and not simply by example, because as we learned in proofs, just because it works for one example does not mean that will be same result for the infinitely many others. I remember in previous math classes having proofs of certain things but never talking about them. I think it is important to find a way to incorporate simple proofs into all mathematics classrooms even in the middle school years. This will only set students up for deeper understanding of mathematical concepts and ideas.
Finally implications for instruction are mentioned (441-5). It is going to be vital to find a way to show all students to see generalizations that can be applied to all cases. In order to help students, the authors employed two means to help students move beyond initial conceptions (441-5), which included, creating tasks that focus student discussion on the validity and power of their justifications and encouraging discussion regarding how students’ justifications related to their generalizations.
Overall, it seems that in order to help students learn to justify we as teachers need to be selective about the types of problems we choose to help students find generalizations and explain why they work. I thought the poster example was a good problem for students to be able to find a way of explaining why their rule would always work. I always think that example helped me to get a better understanding of the differences between the four categories of justification mentioned earlier.
Kaitlin Froehlke
In the article “Why, Why should I justify?” the authors talk about the importance of justification in a mathematics classroom. According to the authors the Beam Design Problem touches four objectives of NCTM standards (3-1). It seems that an important part of justification is convincing others that a statement is valid (3-3). Students can determine if a justification is valid through interacting with their peers and students. These justifications that students come up with fall in to four categories: No justification/procedural justification, empirical justification, generic examples, and deductive justification (3-4). Teachers should aim to get their students to category four, deductive justification. In order for teachers to get students to this category they need to emphasize the importance of justification.
Teachers want students to provide “sound explanations” (4-3). These explanations describe why a formula or model will always work for a certain situation. Another important reason for emphasizing justification is developing understanding that lets others to analyze patterns in different situations (4-4). We as teachers want students to be able to justify their answers when constructing generalizations (like the beam design problem). I think that if teachers are going to emphasize justification, it should be reflected in our grading system, and our assessments of student learning. It seems to me that standardized tests did not really care about my justification, or logic to my answer. Rather most standardized tests just care about the final answer (which ever bubble that needs to be filled in). If a teacher demands that a class justifies their answers they will be better prepared for that teacher’s assessments, and hopefully have a better understanding of how to analyze situations to see patterns and make formulas that are applicable.
For our students to realize the importance of justification in a classroom teachers should encourage discussion in the classroom among students. Teachers need to ask questions that force students to create mathematically valid justifications (5-3). I think that teachers may be able to provide a check list (much like the questions on pg 5-3) that shows students the important steps of creating a valid justification. I think that justification is very important in the middle school classroom. If we get students accustomed to justifying their answers, they will be ahead of many of their peers by the time they get to high school. I think that we as middle school teachers need to emphasize the importance of justification throughout the school year, not just for a few exercises. Developing "good" justification skills can really help our students for the rest of their academic careers and life in general.
Mike Freeland
In their article “Why, Why Should I Justify?” John Lannin, David Barker, and Brian Townsend examine the importance of valid justification in mathematics, why students should explain their reasoning. Before the authors voice their own thoughts, they look at what the NCTM’s principles and standards require of middle school students in this area: that students should move from observing regularities to forming generalizations to evaluating conjectures to constructing mathematical arguments (440-1).
After considering these principles and standards, they assert what constitutes a valid justification. Students’ responses tend to fall into four categories: no justification, experimental justification, generic examples, and deductive justification (440-2). The authors maintain that deductive justification is the best way to convince others why something works mathematically (440-1). That being said, I know from experience that students tend to use the other three types of responses, based on a few examples that they have used. We need to encourage them to deepen their thinking and look beyond a few examples to why a rule works for all cases; middle school students will not naturally come up with a deductive justification on their own.
Next, Lannin, Barker, and Townsend emphasize the importance of justification. They believe students should observe how rules apply to various situations, construct generalizations to similar cases, and reflect on their own reasoning (441-1). Students should be able to explain why their generalizations work for all values of the variable and develop an understanding that allows others to make generalizations for related cases (441-3). Not only should they be able to justify, but they should also be able to evaluate themselves and others regarding the validity of their justification (441-4).
Finally, the authors consider how this idea of justification should impact our classrooms. They encourage teachers to design tasks to center student discussion on justification and reasoning and to encourage discourse about relating justifications and generalizations of students (441-5). They included a list of quality questions to ask students, which encourage them to justify their reasoning (442-2). As in other materials we have read, an important part of creating an atmosphere that encourages justification is establishing and negotiating classroom norms (440-1,2). This leads me to ponder this question: how do we go about creating these norms when it appears to be so much easier for students to settle for less-than-valid justification?
Speaking from a student’s perspective, I often prefer to have the right answer rather than a valid justification. In fact, I have a preoccupation with always wanting to get the right answer; how I obtained it doesn’t matter as much. Thus, it is very frustrating for me personally to thoroughly justify my own solutions in mathematics. Furthermore, this makes it even more difficult to help my students justify, even though I know this is a mathematically beneficial habit for them to develop. A significant issue for many mathematics students, myself included, is that we prefer to possess a superficial understanding of mathematics and rather than being able to truly comprehend and justify. I have decided that I need to improve my justification skills before I will ever be able to help my students. Lannin, Barker, and Townsend offer much-needed insight. By asking myself pointed questions about why a rule works, I force myself to achieve a valid justification and encourage my students to do the same.
Mandi Mills
The article, “Why, Why Should I Justify,” examines the beam problem in terms of students responses and their justifications. The author states that the beam problem was specifically chosen because it covers four activities according to NCTM set of standards and principles. The students should be able to examine patterns and determine regularities, from these regularities the student should be able to formulate generalizations, evaluate conjectures formed, and finally construct and evaluate mathematical expressions based on these conjectures (440-1) . The beam problem accomplishes all of this and allows the teacher to peak into the student’s train of thought behind their idea. Further on in the reading the authors discuss what exactly constitutes a valid mathematical argument. The others come to the conclusion that responses fall into four categories based on their validity. The responses can be categorized as merely being a procedural justification, empirical justification, generic examples, or deductive justification (440-3). It is important discussions around justification take place so that the student can make the connection between the rule and how it works across multiple types of problems (441-2). In the article for example, the same method that was used in the beam problem can be applied to the poster and tack problem. These problems are useful for more than just developing mathematical expressions. The teacher can use these problems with provided solutions and justifications to discuss what constitutes a valid response (442-2). During these discussions it is vital to continue to ask students to explain further and encourage them to do the same when discussing it with peers (442-3). When students know what constitutes a valid argument and are given a chance to compare and assess each other’s arguments their reasoning skills improve. Christopher Cardon
“Why, Why Should I Justify” is an article based on justification in the middle school classroom. The principles and standards for school mathematics calls for middle-grade students to; “Examine patterns and structures to detect regularities, formulate generalizations and conjectures about observed regularities, evaluate conjectures, and construct and evaluate mathematical arguments." 440-1 It is with all these activities where students thought process advance. Students are no longer just making simple justifications where answers can be given by just a simple number; they are explaining what makes an answer what it is. They know how mathematical symbols like 2n change the outcome of a problem. They are providing “valid mathematical explanations for their work” 440-2 which is an advance thought process that not is happening in the elementary age.
Discussions regarding justification allow students to; "Observe how a rule applies across various cases, construct generalizations to related situations, and reflect on their own reasoning regarding the viability of their rules”. 441-2 It is with these situations where students mathematical skills become stronger. They are able to relate mathematical problems to other problems where similar rules may apply and they can catch on to structure to solve the problem. “Another reason for emphasizing justification is that it develops an understanding that permits others to construct generalizations for similar situations”. 441-3 This reason allow students to make generalizations from one idea to the next idea. When students are thinking in this way they can make generalizations for themselves and think about what method works best for them in any circumstance.
Teachers need to be aware of the nctm principles and standards in order to make justifications a norm. “Creating Tasks that focus student discussion on the validity and power of their justification and encouraging discussion regarding how students justification related to their generalization,” 441-6 help students overcome difficulty in constructing justifications. Whole-class discussion help students to validate other students’ conceptions. They are willing to give reason and thought to other students work which focuses on student discussion to validate justifications. They are also withering out the explanations needed in order to make a valid discovery or generalization when solving a problem. My goal as a future educator is to establish a norm in my classroom based on these activities and discussions. It is with these goals I can make my classroom atmosphere a learning one where everyone succeeds .
Fredrick Martin
"Why Should I Justify" focused on the way students found patterns and what was sufficient reasoning for an explanation a student came up with. According to the article "Students determine what constitutes a valid justification from the social interactions they experience with the teacher and other students." (440-4) I agree with the article that it's extremely important to know why rules work, even when the student is the one who comes up with it. It's easy to see an algorithm spit out correct answers, but without knowing why they do what they do will make it harder to remember the algorithm and to understand when to apply them. The article lists quite a few questions to ask students when they're trying to justify their answers. I feel that questions are the best way to go about leading your students to properly justifying their answers. Some of the questions they listed were, "What is changing in this situation?" "What stays the same?" "How do you know your rule will work for 105 posters" (Essentially asking how you can know it will work for some arbitrary amount and not just maybe a few examples.) (442-4) I think one thing I'd change is in not making the questions as specific to the problem. I think trying to make the student relate questions the problem is very important because it forces them to make connections that are otherwise being given to them. Middle schoolers Algebra skills are just starting to be developed, it's a big change from the very basic number calculation that is taught in elementary school (very important tools to be used for algebra but much less conceptual). As the article states: "Alebraic thinking in the middle grades involves constructing generalizations moving beyond the focus of specific calculations in elementary school." This is extremely important to focus on for young children and a time I think a lot of students start to get lost. If children can build a solid foundation in understanding why patterns work then I think many of them can be more successful in their later years in school. ~Doug Wills
In the article “Why, Why Should I Justify?” the authors sought to explain the importance of giving justification to each mathematical problem students encounter. The NCTM says that students should give justifications and expect others to do the same (440-1) and these justifications fall into four categories: (a) no justification/procedural, (b) empirical justification, (c) generic examples, (d) deductive justification (440-3).
I found it interesting that the various students that tried to give justification for how they answered the rod and beam problem had justifications that fell under the four different categories. Student 1 would be very procedural and fits into category (a) (440-3). Student 3 used a generic example like category (c) and didn’t explain how it could be applied to any case (440-4). Student 2 gave a (d) deductive justification which is the most desired and clear explanation (440-5).
Justification is very important because it allows three things to happen for students “(1) observe how a rule applies across various cases, (2) construct generalizations to related situations, and (3) reflect on their own reasoning regarding the viability of their rules,” (441-2). Students should be able to justify their answers to problems and see how they can connect to other similar problems.
I attempted to do the rod and beams problem before reading this article and I came up with a different way to look at the problem than the three students given. My number sentence originally said n + 2n + (n-1) because I broke the levels of the beam up into the bottom, the two middle supports, and the top. I then condensed it into 3n + (n-1) and identified the 3n as the base of the triangle plus the 2 sides then the (n-1) as the top connections.
Tori Ward
POSTED LATE DUE TO TECHNICAL ISSUES.
In the Why, Why Should I Justify? article we see many different examples so they can give their example of the different ways that students, especially middle school students would solve problems for a generic formula. I thought this article was pretty good in terms of its examples however in terms of actually telling us how to make justification happen I found it rather lacking. I feel that Lannin, Barker, and Townsend had an extreme focus on the examples and did not clearly explain the types of justifications. There are four types of justifications (440-3); no justification also known as a procedural justification, empirical justification, generic examples, and the last the deductive justification.
As I understand the difference among the first two and the second two justifications are that the first two aren’t really justifications just explanations of the steps they took to reach the solution or the rule they created to produce a solution. I felt as though just providing examples of the student’s justification weren’t enough I needed the definition to help clarify what should or shouldn’t be declared an acceptable justification. Many reasons or really the only reason the authors gave for why students should have to justify is for students to discuss the relationship between the concrete ideas and the abstract reasoning behind the formula or rule (441-5).
Finally they spend some time on implementation and seemed too focused on asking certain questions in order to justify how a student comes up with his or her response (442-4). I realized as well that in the article whomever the teacher was they had setup norms that allowed this type of environment to flourish. I find that with any classroom we must be completely in control and have the norms to make an environment that is conducive to learning, without the fear of failing (440-4). When the students feel that way then they open up to problem solving and exercises. The beginning of the article also speaks of being able to fulfill the Principles and Standards expectations of what a middle school student should know (440-1).
I know that getting students to connect to the subject may not always be the easiest but justification is one of the necessary ways that we as teachers can expose then to the type of reasoning that is necessary to be successful in mathematics today. It is apparent that the nature of how we do mathematics is changing however the classic principles of what knowledge we need to know and build on has not changed.
Denise Slate
In the article “Why, Why Should I Justify?” authors Lannin, Barker, and Townsend elaborate on the theme of explaining your work as a student. Solid justification leads to a broader understanding of concepts, and formation of generalizations. I agree that strong explanations of solutions are valuable to teachers assessing students’ comprehension, and many students have yet to come across such a challenge. I find myself even struggling explaining how I derive, supposedly trivial, answers.
For those with little experience elaborating on their thoughts or justifying their work, it could take some practice. At points in the article, as well as in our 3500 class, so much time is spent explaining the process to a problem it could take multiple class periods to “solve” one problem. I wonder how reasonable it would be to expect every one of your students to offer, or even form some kind of proof of their knowledge for every concept. The quality of justification this teacher demanded required a lot of time and discussion, and does not emulate (at all!) the classroom I am pre-interning in.I do recognize that my mentor teacher manages her classes differently than I may my own, the article did mention that atmosphere played a huge role in participation in conversations. Atmosphere, or your learning environment, needs to be established and built upon beginning day one. Hopefully that will encourage a more comfortable setting to share ideas.
- Tim H
After reading this article, I realized how important a valid justification is when explaining a problem so that all of the students will understand the problem. Except for my senior year in high school, I never had a math teacher let us work in groups or have a class discussion with a math problem like explained in the article or in the video of class. It was always here is how you do this mathematical idea with some examples that are taken from the examples in the book, and this is what is due for your homework the next class period. We always had to show our work, but it was never a really good valid justification. Maybe some of my work was at level three of generic examples, but for the most part I would say it was empirical justification, and I would always get A’s.
With reflecting back on high school and reading this article I can be able to see how having a class discussion and having a valid justification is very important. I agree with the article that a deductive justification is the best type because it shows a general argument that clearly explains why the rule applies to all cases of the situation, but I think that showing some generic examples with this general case helps with justifying your argument. With the importance of justification, it is very important that the students understand and realized the reasoning on what they are doing (e.g. why am I multiplying by 4?). This is so they are not just manipulating algebraic symbols so that it works and they get the correct answer. Making the students think about what they have done and why it works for this problem, will help them understand other problems in the future and not become so lost when you move on to the next section or problem.
I really liked the questions that they author posted near the end of the article, and how they mentioned that we need to consistently need to be asking the students these questions. I agree with them that this will encourage the students to start thinking of these questions on their own and towards each other. By asking these questions, I believe that it will help students understand how to make a valid justification for their solution on how to solve questions, and as the author mentioned, create a whole-class discussion which is what we want all of the students to do.
Bret Van Zanten
The readings reflections have two main purposes:
1) to hold you accountable for careful reading of and reflection on the readings assigned in class; and
2) to provide you with a record of what you've learned and thought about as a result of the readings.
The readings reflections will be evaluated using the following criteria:
- completeness and timeliness of the entries;
- comprehension of the main ideas of the readings; and
- depth and quality of integration of the ideas with your own thinking.
Submit your readings reflection before reading anyone's on the Wiki page and then paste it into the existing reflection page for that current reading. This is due by Sun 10/9 by 11pm.The article, "Why, Why Should I Justify?", discusses different student approaches for the Beam Design problem, which incoporates four NCTM Principles and Standards. It can be challenging for students to justify their answers, so it is up to the teacher to create an environment condusive to these types of problems. I believe the teacher has to establish these norms early on in the school year. The longer students go without justifying their answers, the harder it is going to be for them to justify later in life. I am a product of this. For most of my schooling, I was never required to justify my generalizations, now it is difficult to get in that mindset. In the article, it mentions starting as early as fifth grade (440-3). So as middle school teachers, we should strongly encourage our students to validate their generalizations.
The four categories of justification are as follows: no justification/procedural justification, empirical justification, generic examples, and deductive justification. The goal is for students to perform deductive justification, which means to show how the rule was derived and be able to explain how it works. Proper justifications will apply to all cases, not just generic examples. Requiring justification in class also requires students to be confident. Students need their generalizations to work for all cases. Thus, they may need to try more than one approach to get a justification that works. If a student lacks mathematical confidence, they will be less likely to take the time to find a generalization that applies to all cases. This also must be encouraged at a young grade.
Students that are able to justify not only help themselves, but help their peers to understand the situation. Teachers need to encourage justification discussions and tasks that focus these discussions. The more we implement activities, such as the Beam Design, the better off our students will be. They will develop problem solving skills that will assist them not just in math class, but in the real world.
Hailey McDonell
The article “Why, Why Should I Justify?” explored the idea of students learning how to justify the solutions they are coming up with. As I read the article, I couldn’t help but notice how much I identified with the students at times. At the same time, the role of the teacher discussed and the way the classroom should be run seems to not only be practical but also quite beneficial for the students.
Justification is pertinent to explaining how you came to a solution because it provides validity to your solution. Additionally, it helps convince others your explanation is valid (3-2). A weak justification, or none at all, leaves those who are trying to see how you obtained a certain solution confused and not convinced. The article stated there four types of broad justifications: (1) no justification or procedural only, (2) empirical, (3) generic examples, and (4) deductive justification (3-3). Many times, I believe, teachers allow their students to believe procedural justification is okay and works as means of an explanation. As we saw when we saw when we watched Cathy’s video, one student’s justification was “flip the fraction and multiply”. This does not justify anything mathematically. This may be the step the student takes to get to an answer but that doesn’t mean it proves anything. Additionally, from personal experience, students believe examples prove it all. I know when I was in the proofs course last semester it was engrained in my head to not assume if it worked for some it worked for all. There may always be that counter example. However, coming from a middle school mind, I can see how examples may seem like justification. It seems to be showing why something works. It is our job as teachers to show them why it is not justification. We can do this by asking the right questions.
The questions asked by the teachers seem to be most important piece in changing the student’s way of thinking about justification. There are two things teachers do to help students think in the way of correct justification: (1) creating tasks which focus on justification and (2) encouraging justification through discussion and questions (4-5). As we have read before, asking questions forces the students to think for themselves. It puts the thinking back on them. I know as someone who is currently a student, when a teacher lectures and tells answers, acknowledges my answer as correct, I shut down. I know this is true for many if not all other students. If I know I have the right answer, what matters what the reasoning is. It seems to be the role of the teacher to make this different.
The idea of group discussion and connecting the different techniques of doing a problem seems to be something that is very beneficial to students. First off the can see how others approached the problem and, hopefully, be able to connect their own ideas to that of others. More importantly, they will be able to test the validity of their own as well as others’ justification (6-1). This is a way for them to test if they know and have effective reasoning. This is something I feel I didn’t have the chance to do very much when I was in school and I feel if I did, I would have been able to examine more closely the reasoning behind my own work in school.
Katey Cook
As students make the leap from concrete numerical procedures to more abstract algebra with variables, we, as teachers, must continue to push our students to justify their reasoning behind their mathematical ideas. These initial algebraic problems, like the Beam Design task, may look intimidating because of the new concept of a variable, but they are essential in bridging the gap between concrete and abstract thinking. By asking for justification of simple algebraic models, we allow students to see how a rule applies across various cases, construct generalizations to similar situations, and reflect on their own reasoning of their rules (4-2). This will promote the recognition of patterns for our students, who will become more adept at creating algebraic rules as they learn to assess their own reasoning.
As teachers, we must recognize which kind of justification is sufficient and guide students to provide this kind of justification for their ideas. Lannin, Barker, and Townsend classify justification using the subgroups of Simon and Blume (3-4). They are: (1) no/procedural justification, this would be characterized by statements like “I don’t know, just because,” as the student is unable to provide answers to why or how the rule works (2) Empirical justification, which would be the justification a student has if he or she is able to consider multiple cases, but has not provided reasoning for why the rule works over all of n (3) Generic examples are those in which students choose n to be some number, and explain their rule in terms of this one variable rather than in terms of any n (4) Deductive justification, this is the kind of justification that we must promote in our classrooms. Deductive justification explains why and how the rule works for all of n. These are the kinds of justifications that convince the classroom that the student’s statement is valid.
Though a student who provides a (1) justification may be aware that he or she is in need of help, students who provide type (2) and (3) justifications may feel more confident. They perhaps feel that they have reasoned out the answer because it works for random cases because they are more comfortable thinking in terms of concrete numbers rather than abstract variables. With these students, I feel that it is important to let them know that testing one’s rule on sample cases is a good idea to check one’s answer, but it is not how we can create a rule that works for all of n. This is why the article encourages questions from the teacher like, “How do you know your rule will always work?” and “How does your rule relate to the problem situation?” (5-3). These types of focusing questions will get students thinking in terms of any variable n rather than in terms of specific cases, n=5, n=8, etc.
Though the subject matter has changed from concrete numbers to abstract variables, it is important that we, as teachers, demand justification from our students so that they can continue to expand their reasoning capacities.
Valerie Gipper
The article, “Why, Why Should I Justify?”, discusses the importance of teaching students to not just conclude that an equation works for a variety of examples, but why the equation makes sense. The article goes on to discuss the four categories that student justifications generally fall into (440-4). We then see the three students explanations from the beginning of the article explained with regards to these four categories (440- 4). It seems that the fourth category (deductive justification) is the highest level of justification. In this level of justification the students reasoning applies to all situations and not just specific examples like in the other three cases (440-7).
Next the article talks about why teaching students to justify is important (441-1), which includes three reasons. I think it is very important for students to learn how to justify why things work and not simply by example, because as we learned in proofs, just because it works for one example does not mean that will be same result for the infinitely many others. I remember in previous math classes having proofs of certain things but never talking about them. I think it is important to find a way to incorporate simple proofs into all mathematics classrooms even in the middle school years. This will only set students up for deeper understanding of mathematical concepts and ideas.
Finally implications for instruction are mentioned (441-5). It is going to be vital to find a way to show all students to see generalizations that can be applied to all cases. In order to help students, the authors employed two means to help students move beyond initial conceptions (441-5), which included, creating tasks that focus student discussion on the validity and power of their justifications and encouraging discussion regarding how students’ justifications related to their generalizations.
Overall, it seems that in order to help students learn to justify we as teachers need to be selective about the types of problems we choose to help students find generalizations and explain why they work. I thought the poster example was a good problem for students to be able to find a way of explaining why their rule would always work. I always think that example helped me to get a better understanding of the differences between the four categories of justification mentioned earlier.
Kaitlin Froehlke
In the article “Why, Why should I justify?” the authors talk about the importance of justification in a mathematics classroom. According to the authors the Beam Design Problem touches four objectives of NCTM standards (3-1). It seems that an important part of justification is convincing others that a statement is valid (3-3). Students can determine if a justification is valid through interacting with their peers and students. These justifications that students come up with fall in to four categories: No justification/procedural justification, empirical justification, generic examples, and deductive justification (3-4). Teachers should aim to get their students to category four, deductive justification. In order for teachers to get students to this category they need to emphasize the importance of justification.
Teachers want students to provide “sound explanations” (4-3). These explanations describe why a formula or model will always work for a certain situation. Another important reason for emphasizing justification is developing understanding that lets others to analyze patterns in different situations (4-4). We as teachers want students to be able to justify their answers when constructing generalizations (like the beam design problem). I think that if teachers are going to emphasize justification, it should be reflected in our grading system, and our assessments of student learning. It seems to me that standardized tests did not really care about my justification, or logic to my answer. Rather most standardized tests just care about the final answer (which ever bubble that needs to be filled in). If a teacher demands that a class justifies their answers they will be better prepared for that teacher’s assessments, and hopefully have a better understanding of how to analyze situations to see patterns and make formulas that are applicable.
For our students to realize the importance of justification in a classroom teachers should encourage discussion in the classroom among students. Teachers need to ask questions that force students to create mathematically valid justifications (5-3). I think that teachers may be able to provide a check list (much like the questions on pg 5-3) that shows students the important steps of creating a valid justification. I think that justification is very important in the middle school classroom. If we get students accustomed to justifying their answers, they will be ahead of many of their peers by the time they get to high school. I think that we as middle school teachers need to emphasize the importance of justification throughout the school year, not just for a few exercises. Developing "good" justification skills can really help our students for the rest of their academic careers and life in general.
Mike Freeland
In their article “Why, Why Should I Justify?” John Lannin, David Barker, and Brian Townsend examine the importance of valid justification in mathematics, why students should explain their reasoning. Before the authors voice their own thoughts, they look at what the NCTM’s principles and standards require of middle school students in this area: that students should move from observing regularities to forming generalizations to evaluating conjectures to constructing mathematical arguments (440-1).
After considering these principles and standards, they assert what constitutes a valid justification. Students’ responses tend to fall into four categories: no justification, experimental justification, generic examples, and deductive justification (440-2). The authors maintain that deductive justification is the best way to convince others why something works mathematically (440-1). That being said, I know from experience that students tend to use the other three types of responses, based on a few examples that they have used. We need to encourage them to deepen their thinking and look beyond a few examples to why a rule works for all cases; middle school students will not naturally come up with a deductive justification on their own.
Next, Lannin, Barker, and Townsend emphasize the importance of justification. They believe students should observe how rules apply to various situations, construct generalizations to similar cases, and reflect on their own reasoning (441-1). Students should be able to explain why their generalizations work for all values of the variable and develop an understanding that allows others to make generalizations for related cases (441-3). Not only should they be able to justify, but they should also be able to evaluate themselves and others regarding the validity of their justification (441-4).
Finally, the authors consider how this idea of justification should impact our classrooms. They encourage teachers to design tasks to center student discussion on justification and reasoning and to encourage discourse about relating justifications and generalizations of students (441-5). They included a list of quality questions to ask students, which encourage them to justify their reasoning (442-2). As in other materials we have read, an important part of creating an atmosphere that encourages justification is establishing and negotiating classroom norms (440-1,2). This leads me to ponder this question: how do we go about creating these norms when it appears to be so much easier for students to settle for less-than-valid justification?
Speaking from a student’s perspective, I often prefer to have the right answer rather than a valid justification. In fact, I have a preoccupation with always wanting to get the right answer; how I obtained it doesn’t matter as much. Thus, it is very frustrating for me personally to thoroughly justify my own solutions in mathematics. Furthermore, this makes it even more difficult to help my students justify, even though I know this is a mathematically beneficial habit for them to develop. A significant issue for many mathematics students, myself included, is that we prefer to possess a superficial understanding of mathematics and rather than being able to truly comprehend and justify. I have decided that I need to improve my justification skills before I will ever be able to help my students. Lannin, Barker, and Townsend offer much-needed insight. By asking myself pointed questions about why a rule works, I force myself to achieve a valid justification and encourage my students to do the same.
Mandi Mills
The article, “Why, Why Should I Justify,” examines the beam problem in terms of students responses and their justifications. The author states that the beam problem was specifically chosen because it covers four activities according to NCTM set of standards and principles. The students should be able to examine patterns and determine regularities, from these regularities the student should be able to formulate generalizations, evaluate conjectures formed, and finally construct and evaluate mathematical expressions based on these conjectures (440-1) . The beam problem accomplishes all of this and allows the teacher to peak into the student’s train of thought behind their idea.
Further on in the reading the authors discuss what exactly constitutes a valid mathematical argument. The others come to the conclusion that responses fall into four categories based on their validity. The responses can be categorized as merely being a procedural justification, empirical justification, generic examples, or deductive justification (440-3). It is important discussions around justification take place so that the student can make the connection between the rule and how it works across multiple types of problems (441-2). In the article for example, the same method that was used in the beam problem can be applied to the poster and tack problem.
These problems are useful for more than just developing mathematical expressions. The teacher can use these problems with provided solutions and justifications to discuss what constitutes a valid response (442-2). During these discussions it is vital to continue to ask students to explain further and encourage them to do the same when discussing it with peers (442-3). When students know what constitutes a valid argument and are given a chance to compare and assess each other’s arguments their reasoning skills improve.
Christopher Cardon
“Why, Why Should I Justify” is an article based on justification in the middle school classroom. The principles and standards for school mathematics calls for middle-grade students to; “Examine patterns and structures to detect regularities, formulate generalizations and conjectures about observed regularities, evaluate conjectures, and construct and evaluate mathematical arguments." 440-1 It is with all these activities where students thought process advance. Students are no longer just making simple justifications where answers can be given by just a simple number; they are explaining what makes an answer what it is. They know how mathematical symbols like 2n change the outcome of a problem. They are providing “valid mathematical explanations for their work” 440-2 which is an advance thought process that not is happening in the elementary age.
Discussions regarding justification allow students to; "Observe how a rule applies across various cases, construct generalizations to related situations, and reflect on their own reasoning regarding the viability of their rules”. 441-2 It is with these situations where students mathematical skills become stronger. They are able to relate mathematical problems to other problems where similar rules may apply and they can catch on to structure to solve the problem. “Another reason for emphasizing justification is that it develops an understanding that permits others to construct generalizations for similar situations”. 441-3 This reason allow students to make generalizations from one idea to the next idea. When students are thinking in this way they can make generalizations for themselves and think about what method works best for them in any circumstance.
Teachers need to be aware of the nctm principles and standards in order to make justifications a norm. “Creating Tasks that focus student discussion on the validity and power of their justification and encouraging discussion regarding how students justification related to their generalization,” 441-6 help students overcome difficulty in constructing justifications. Whole-class discussion help students to validate other students’ conceptions. They are willing to give reason and thought to other students work which focuses on student discussion to validate justifications. They are also withering out the explanations needed in order to make a valid discovery or generalization when solving a problem. My goal as a future educator is to establish a norm in my classroom based on these activities and discussions. It is with these goals I can make my classroom atmosphere a learning one where everyone succeeds .
Fredrick Martin
"Why Should I Justify" focused on the way students found patterns and what was sufficient reasoning for an explanation a student came up with. According to the article "Students determine what constitutes a valid justification from the social interactions they experience with the teacher and other students." (440-4) I agree with the article that it's extremely important to know why rules work, even when the student is the one who comes up with it. It's easy to see an algorithm spit out correct answers, but without knowing why they do what they do will make it harder to remember the algorithm and to understand when to apply them.
The article lists quite a few questions to ask students when they're trying to justify their answers. I feel that questions are the best way to go about leading your students to properly justifying their answers. Some of the questions they listed were, "What is changing in this situation?" "What stays the same?" "How do you know your rule will work for 105 posters" (Essentially asking how you can know it will work for some arbitrary amount and not just maybe a few examples.) (442-4) I think one thing I'd change is in not making the questions as specific to the problem. I think trying to make the student relate questions the problem is very important because it forces them to make connections that are otherwise being given to them.
Middle schoolers Algebra skills are just starting to be developed, it's a big change from the very basic number calculation that is taught in elementary school (very important tools to be used for algebra but much less conceptual). As the article states: "Alebraic thinking in the middle grades involves constructing generalizations moving beyond the focus of specific calculations in elementary school." This is extremely important to focus on for young children and a time I think a lot of students start to get lost. If children can build a solid foundation in understanding why patterns work then I think many of them can be more successful in their later years in school.
~Doug Wills
In the article “Why, Why Should I Justify?” the authors sought to explain the importance of giving justification to each mathematical problem students encounter. The NCTM says that students should give justifications and expect others to do the same (440-1) and these justifications fall into four categories: (a) no justification/procedural, (b) empirical justification, (c) generic examples, (d) deductive justification (440-3).
I found it interesting that the various students that tried to give justification for how they answered the rod and beam problem had justifications that fell under the four different categories. Student 1 would be very procedural and fits into category (a) (440-3). Student 3 used a generic example like category (c) and didn’t explain how it could be applied to any case (440-4). Student 2 gave a (d) deductive justification which is the most desired and clear explanation (440-5).
Justification is very important because it allows three things to happen for students “(1) observe how a rule applies across various cases, (2) construct generalizations to related situations, and (3) reflect on their own reasoning regarding the viability of their rules,” (441-2). Students should be able to justify their answers to problems and see how they can connect to other similar problems.
I attempted to do the rod and beams problem before reading this article and I came up with a different way to look at the problem than the three students given. My number sentence originally said n + 2n + (n-1) because I broke the levels of the beam up into the bottom, the two middle supports, and the top. I then condensed it into 3n + (n-1) and identified the 3n as the base of the triangle plus the 2 sides then the (n-1) as the top connections.
Tori Ward
POSTED LATE DUE TO TECHNICAL ISSUES.
In the Why, Why Should I Justify? article we see many different examples so they can give their example of the different ways that students, especially middle school students would solve problems for a generic formula. I thought this article was pretty good in terms of its examples however in terms of actually telling us how to make justification happen I found it rather lacking. I feel that Lannin, Barker, and Townsend had an extreme focus on the examples and did not clearly explain the types of justifications. There are four types of justifications (440-3); no justification also known as a procedural justification, empirical justification, generic examples, and the last the deductive justification.
As I understand the difference among the first two and the second two justifications are that the first two aren’t really justifications just explanations of the steps they took to reach the solution or the rule they created to produce a solution. I felt as though just providing examples of the student’s justification weren’t enough I needed the definition to help clarify what should or shouldn’t be declared an acceptable justification. Many reasons or really the only reason the authors gave for why students should have to justify is for students to discuss the relationship between the concrete ideas and the abstract reasoning behind the formula or rule (441-5).
Finally they spend some time on implementation and seemed too focused on asking certain questions in order to justify how a student comes up with his or her response (442-4). I realized as well that in the article whomever the teacher was they had setup norms that allowed this type of environment to flourish. I find that with any classroom we must be completely in control and have the norms to make an environment that is conducive to learning, without the fear of failing (440-4). When the students feel that way then they open up to problem solving and exercises. The beginning of the article also speaks of being able to fulfill the Principles and Standards expectations of what a middle school student should know (440-1).
I know that getting students to connect to the subject may not always be the easiest but justification is one of the necessary ways that we as teachers can expose then to the type of reasoning that is necessary to be successful in mathematics today. It is apparent that the nature of how we do mathematics is changing however the classic principles of what knowledge we need to know and build on has not changed.
Denise Slate
In the article “Why, Why Should I Justify?” authors Lannin, Barker, and Townsend elaborate on the theme of explaining your work as a student. Solid justification leads to a broader understanding of concepts, and formation of generalizations. I agree that strong explanations of solutions are valuable to teachers assessing students’ comprehension, and many students have yet to come across such a challenge. I find myself even struggling explaining how I derive, supposedly trivial, answers.
For those with little experience elaborating on their thoughts or justifying their work, it could take some practice. At points in the article, as well as in our 3500 class, so much time is spent explaining the process to a problem it could take multiple class periods to “solve” one problem. I wonder how reasonable it would be to expect every one of your students to offer, or even form some kind of proof of their knowledge for every concept. The quality of justification this teacher demanded required a lot of time and discussion, and does not emulate (at all!) the classroom I am pre-interning in.I do recognize that my mentor teacher manages her classes differently than I may my own, the article did mention that atmosphere played a huge role in participation in conversations. Atmosphere, or your learning environment, needs to be established and built upon beginning day one. Hopefully that will encourage a more comfortable setting to share ideas.
- Tim H
After reading this article, I realized how important a valid justification is when explaining a problem so that all of the students will understand the problem. Except for my senior year in high school, I never had a math teacher let us work in groups or have a class discussion with a math problem like explained in the article or in the video of class. It was always here is how you do this mathematical idea with some examples that are taken from the examples in the book, and this is what is due for your homework the next class period. We always had to show our work, but it was never a really good valid justification. Maybe some of my work was at level three of generic examples, but for the most part I would say it was empirical justification, and I would always get A’s.
With reflecting back on high school and reading this article I can be able to see how having a class discussion and having a valid justification is very important. I agree with the article that a deductive justification is the best type because it shows a general argument that clearly explains why the rule applies to all cases of the situation, but I think that showing some generic examples with this general case helps with justifying your argument. With the importance of justification, it is very important that the students understand and realized the reasoning on what they are doing (e.g. why am I multiplying by 4?). This is so they are not just manipulating algebraic symbols so that it works and they get the correct answer. Making the students think about what they have done and why it works for this problem, will help them understand other problems in the future and not become so lost when you move on to the next section or problem.
I really liked the questions that they author posted near the end of the article, and how they mentioned that we need to consistently need to be asking the students these questions. I agree with them that this will encourage the students to start thinking of these questions on their own and towards each other. By asking these questions, I believe that it will help students understand how to make a valid justification for their solution on how to solve questions, and as the author mentioned, create a whole-class discussion which is what we want all of the students to do.
Bret Van Zanten